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Capacitive Reactance of an Electrical Circuit :
3. - ELECTRICAL REACTANCE
The circuits considered so far are said resistive or ohmic because they include exclusively resistors.
At present, we will be interested in capacitive and inductive circuits that will be "attacked" by a generator of sinusoidal alternating quantities.
Capacitive and inductive circuits are called reactive circuits because they have no heat effects but have other properties that we will examine elsewhere.
3. 1. - CAPACITIVE CIRCUITS
In Figure 5 is shown the simplest capacitive circuit which comprises only one capacitor which is applied a DC voltage (Figure 5-a) or alternative (Figure 5-b).
In the case of a group of capacitors, in series or in parallel, it will simplify the assembly and bring the grouping to a single capacitor by the formulas seen in previous lessons.
We know that in the circuit of Figure 5-a only the current necessary to charge the capacitor as soon as this element is connected to the battery is circulated, and that, in this same circuit, all circulation of the current ceases when the capacitor has been charged at the same voltage as that of the battery : we can say that a capacitor prevents the passage of a direct current.
A current is again obtained in the circuit when the battery is disconnected and a resistor is substituted in which the capacitor discharge current is then passed.
It can therefore be seen that, in a capacitive circuit, the current flows only when the voltage applied to the capacitor plates varies : in fact, when the capacitor is connected to the battery, and the voltage between its armatures therefore increases by value provided by the battery, we have in the circuit a charging current while we have a discharge current when the same voltage decreases to zero.
It is therefore understood that, if a sinusoidal variable voltage, such as that provided by the generator of Figure 5-b, is applied, a current will flow at each moment due to the successive charges and discharges of the capacitor. Since the capacitor plates are directly connected to the poles of the generator, there must be between them at each moment, the same voltage as that provided by the generator.
Therefore, when increasing the voltage supplied by the generator, the capacitor is charged so that between its armatures there is at each moment a voltage equal to that which the generator provides progressively : in the circuit therefore circulates the charge current of the capacitor.
When, on the contrary, the voltage supplied by the generator decreases, the capacitor is discharged so that, in this case also there is at each moment between the two armatures a voltage equal to that of the generator : in the circuit circulates now the discharge current of the capacitor.
It is thus demonstrated that the capacitive circuit is traversed by the charging current when the voltage increases, while it is traversed by the discharge current when the voltage decreases.
Let us now see what form the current, if we assume that the generator applies to the capacitor the alternating voltage represented by the graph of Figure 6-c, where are indicated by a strong line the parts of the sinusoid which correspond to the increase of the voltage, to distinguish them from the parts which correspond to the diminution of this tension.
It could be shown that if the voltage is sinusoidal, so is the current, as we have already seen in the case of resistances ; but for capacitors, the sinusoid which represents the current is displaced with respect to that which represents the voltage, as we will see.
It is deduced from Figure 6-c that, during the time interval between the instants t = 0 seconds and t = 0.05 seconds, the voltage is positive and increases by passing from the value of 0 V to the maximum value During this time, the capacitor is charged by a current directed from the positive terminal of the generator to the negative terminal, as indicated by the arrows in Figure 6-a.
During the time interval between the instants t = 0.05 seconds and t = 0.1 s, the voltage is still positive but decreases, from the maximum value to the zero value.
Consequently, the capacitor discharges thanks to the current which must flow in the opposite direction to that of the preceding one, that is to say as indicated by the arrows of Figure 6-b, since all the charges which had passed from armatures to the other during the previous charge must turn back during the discharge, so that between the armatures, there is no more voltage at the time of 0.1 seconds when it also vanishes.
At time t = 0.1 s, the voltage is again zero and the generator reverses its polarities ; that is why during the time interval between the instants t = 0.1 seconds and t = 0.15 s, the voltage is negative and increases by passing from zero to the maximum negative value. Therefore, the capacitor is recharged by the current that is directed, in this case also, from the positive terminal of the generator to its negative terminal, as indicated by the arrows in Figure 6-d. Since the generator has changed its polarities, this current flows in the opposite direction to that of the previous load (Figure 6-a).
After reaching the maximum negative value, the voltage re-decreases in the time interval between the instants t = 0.15 seconds and t = 0.2 seconds to cancel at time t = 0.2 s.
During this time, the discharge of the capacitor is again thanks to the directed current, in this case also, in the opposite direction to the previous charging current, as shown by the arrows of Figure 6-e. Always because of the inversion of the polarity of the generator, this current also flows in the opposite direction to that of the previous discharge (Figure 6-b).
To graphically represent the shape of the current flowing in the capacitive circuit, we observe above all that this current must be canceled at the instants to which the reversal of its flow direction corresponds : let us therefore study Figure 6 to see how this happens. (Diagram shown below).
As long as the voltage is positive and increases (between 0 s and 0.05 s), the current flows in the direction indicated in Figure 6-a, whereas when the voltage, still positive, decreases (between 0.05 s and 0.1 s), the current flows in the opposite direction, as shown in Figure 6-b. It is obvious that the current changes its direction of circulation and therefore vanishes when the voltage ceases to increase and is ready to decrease, that is to say when it reaches its maximum value, which corresponds to the time of 0.05 seconds.
We can repeat the same reasoning for the negative half-period of tension ; Referring to Figure 6-d and Figure 6-e, we notice that the current is canceled when the voltage reaches the maximum negative value, that is to say that which corresponds to the time of 0,15 seconds.
We know that the sinusoid which represents the shape of the current must cut the horizontal axis at times of 0.05 seconds and 0.15 seconds ; but to trace this sinusoid, we still have to see what happens when the current values different from zero are positive or negative, to know if we should carry them above or below the horizontal axis.
For that, let us remember that previously, for the resistances, we have already decided to consider the positive current when it left the pole of the generator designated by the letter A, and negative, when on the contrary it entered by this same pole.
If we stick to this convention, we find that between 0 second and 0.05 second the current is positive because it leaves the pole designated by A, as in Figure 6-a ; on the contrary, between 0.05 seconds and 0.1 seconds, as between 0.1 seconds and 0.15 seconds, the current is negative because it enters through the pole A, as in Figures 6-b and 6-d ; finally, between 0.15 seconds and 0.2 seconds, the current is again positive because it leaves the pole A, as in Figure 6-e.
So, if we draw the sinusoid above the horizontal axis when the current is positive, and below this axis when it is negative, and if we also take into account that this current is for the times of 0.15 seconds and 0.05 seconds, the curve of Figure 7, which represents the shape of the current flowing in the capacitive circuit, in the case where this current has the maximum value of, is obtained, 1.5 ampere.
It immediately appears that this curve is different from the sinusoids studied so far, for example from that of Figure 6-c, which represents the shape of the voltage : this is due to the fact already emphasized above, that the sinusoid representing the current is displaced in relation to the tension, unlike what happens in the case of resistance.
To see clearly what this difference is, refer to Figure 8.
In Figure 8-a is shown the AC voltage ; its two cycles are represented by two sinusoids, the second being drawn in strong lines to distinguish it clearly from the first.
In Figure 8-b, on the contrary, we see the shape of the current that the voltage just mentioned makes circulate in a resistor : it is clearly seen that for each sinusoid representing a cycle of the voltage corresponds a similar sinusoid representing a cycle current.
This means that, in the case of a resistor, the voltage and the current vary in concordance, that is to say that they reach the maximum values and the null values at the same times : it is said that for a resistance voltage and current are in phase.
Finally, consider Figure 8-c, on which is represented the shape of the current that a voltage identical to that of Figure 8-a circulates in a capacitive circuit ; we observe first of all that, in each of the two cycles, the same curve as in Figure 7 is repeated.
Then we observe that in Figure 8-c, we also find the sinusoid drawn in strong line, analogous to that of the two upper figures ; which means that in the capacitive circuit the current also has a sinusoidal shape, as already mentioned above.
In this case, however, the sinusoid is shifted to the left by a quarter of a period : indeed, while the strong-line sinusoids of Figures 8-a and 8-b start at 0.2 seconds and end at 0,4 seconds, the sinusoid in the strong line of Figure 8-c starts at 0.15 seconds and ends at 0.35 seconds, that is to say before the others; since the period is 0.2 seconds, the time of 0.05 seconds corresponds to a quarter of a period (0.2 / 4 = 0.05).
As a result of the shift to the left, the sinusoid drawn in a finer line is incomplete ; it misses the part drawn in dotted line on the left of the vertical axis.
We thus see that, in a capacitive circuit, the current has the same form as the voltage, but that each variation of it takes place a quarter of a period before the identical variation of the tension ; the maximum values and the zero values are thus reached by the current with a quarter-period advance with respect to the voltage.
Unlike what happens for the resistors, the current and the voltage are no longer in phase, and it is said that between these two quantities there is a phase shift.
Since the current is a quarter of a period longer than the voltage, we can also say that, in a capacitive circuit, the current is out of phase by a quarter of a period with respect to the voltage.
From all this it is interesting to know not only the form of the voltage and the alternating current, but also the phase difference which exists between these magnitudes ; this is why, in addition to the graphical representation, we also use another type of representation that allows us to immediately see the phase shift between the alternative magnitudes : the vector representation.
We find in Figure 9 the two vectors that indicate the current and voltage : that of the current is vertical because at the beginning of the cycle its value is maximum while that of the voltage is horizontal because its value is zero.
These two vectors form between them an angle of 90° called angle of phase shift.
It should be remembered that the phase shift is often indicated by the corresponding angle, for example, by saying that the phase shift is 90°, instead of saying that it is a quarter of a period.
We saw that the current circulated in a capacitive circuit fed by an AC voltage, determined the shape of this current, and found a system of representation able to highlight the phase difference which exists between the current and the tension. We now only have to see from which elements the intensity of the current obtained depends, by applying to a capacitive circuit a determined alternating voltage.
But let us first remember that this current is due to the successive charges and discharges of the capacitor, and that, therefore, the higher its capacitance, the more the current necessary to charge it to the same voltage as the generator is intense.
We can say that the current flowing in a capacitive circuit is all the more intense as the capacity of the capacitor is greater.
We note here a notable difference between the behavior of a capacitor and that of a resistor : indeed, in the case of resistance, the current is less intense as the resistance is greater ; in the case of a capacitor, on the contrary, the current is all the more intense as the capacitance is greater.
It follows from all this that the voltage required to pass a determined current in a resistor is obtained by multiplying this current by the value of the resistive element and that in the case of a pure capacitive circuit, divide this current by the capacity of the capacitor. In the latter case, however, the division of the current by the capacitance does not give the voltage, because it must also be taken into account that the resistive element always opposes the same resistance to any type of current while a capacitor prevents the circulation of a direct current and more or less allows the circulation of an alternating current according to its frequency.
So, one must not divide the current only by the capacity but also by the pulsation ( ) which precisely indicates the speed with which the tension varies ; it is related to the number of cycles completed in one second, that is to say to the frequency.
The magnitude defined by the ratio V / I in the case of the capacitor is called capacitive reactance, it is indicated by the symbol Xc and measured in Ohms.
Consequently, the resistance that the current encounters when crossing the circuit, that is to say the reactance Xc offered by the capacitor, will be weaker as the capacitance is high and the pulsation is great ; we express Xc by the following relation :
So we see that, for a circuit that includes capacitors, the capacitive reactance is the equivalent of the resistance for a circuit that includes resistive elements ; this means, in other words, that if a resistor opposes the passage of the current by offering a certain resistivity, a capacitor also reacts to the flow of the current by offering a reactance.
It is therefore also possible to apply the OHM law to the capacitive circuit, provided that the resistance is replaced by the reactance presented by the circuit ; we then obtain :
V = Xc . I
Where V and I represent the effective values of voltage and current.
To best use these formulas, let's take an example : What is the voltage drop produced across a capacitor C of 10 nF traversed by an alternating current (I) of 20 mA at the frequency of 100 kHz ?
We have seen the sinusoidal behavior of the resistive and capacitive circuits, let us now see that of the inductive circuit.
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